Igor Vlasenko, Sergiy Maksymenko
This paper studies one-dimensional non-Hausdorff manifolds that are similar to "graphs with split vertices". It is shown that if $M$ is a connected one-dimensional non-Hausdorff manifold such that the set of its "non-Hausdorff" points is locally finite, and each component of its complement has a countable base, then there exists a quotient map $π\colon M \to Γ$ onto an open one-dimensional CW complex, which maps the non-Hausdorff points of $M$ to the vertices of $Γ$. Moreover, $Γ$ is the minimal Hausdorff quotient of $M$, that is, for every continuous map $f\colon M \to N$ into a Hausdorff space $N$, there exists a unique continuous map $\hat{f}\colon Γ\to N$ such that $f = \hat{f} \circ π$.
Pierros Ntelis
This article presents a novel mathematical formalism for advanced manifold--metric pairs, enhancing the frameworks of geometry and topology. We construct various D-dimensional manifolds and their associated metric spaces using functional methods, with a focus on integrating concepts from mathematical physics, field theory, topology, algebra, probability, and statistics. Our methodology employs rigorous mathematical construction proofs and logical foundations to develop generalized manifold--metric pairs, including homogeneous and isotropic expanding manifolds, as well as probabilistic and entropic variants. Key results include the establishment of metrizability for topological manifolds via the Urysohn Metrization Theorem, the formulation of higher-rank tensor metrics, and the exploration of complex and quaternionic codomains with applications to cosmological models like the expanding spacetime. By combining spacetime generalized sets with information-theoretic and probabilistic approaches, we achieve a unified framework that advances the understanding of manifold--metric interactions and their physical implications.
Mihai Turinici
The 2015 fixed point result on rs-relational metric spaces due to Alam and Imdad [J. Fixed Point Th. Appl., 17 (2015), 693-702] is equivalent with the classical Banach Contraction Principle [Fund. Math., 3 (1922), 133-181]. This is also valid for the 1961 statement in metric spaces due to Edelstein [Proc. Amer. Math. Soc., 12 (1961), 7-10], or the 2005 fixed point result in quasi-ordered metric spaces obtained by Nieto and Rodriguez-Lopez [Order, 22 (2005), 223-239].
Tattwamasi Amrutam, Yongle Jiang
We introduce the notion of confined subalgebras in the context of the group von Neumann algebra. We also define Uniformly Recurrent States -- an operator-algebraic analog of Uniformly Recurrent Subgroups. Using this framework, we show that a countable discrete group is $C^*$-simple if and only if it admits no non-trivial amenable confined subalgebras. This generalizes the well-known result of Kennedy that characterizes $C^*$-simplicity in terms of trivial amenable uniformly recurrent subgroups.
Oleksiy Dovgoshey, Olga Rovenska
Let $G=(V,E)$ be a finite connected graph with vertex set $V$ and edge set $E$, and let $U(G)$ be the set of all ultrametric spaces $(V,d_l)$ generated by vertex labelings $l\colon V \to \mathbb R^+$. We prove that the inequality $$ |D(V)| \le |E| + 1 $$ holds for all $(V,d_l) \in U(G)$, where $D(V)$ is the distance set of $(V,d_l)$. The necessary and sufficient conditions under which the above inequality turns to an equality are found. Moreover, we prove that each connected graph with non-negative vertex labeling generates a pseudoultrametric space and find some sufficient conditions under which this space is ultrametric.
Leandro Aurichi, Gustavo Boska, Davide Giacopello, Paulo Magalhães Júnior
In 1992, Diestel asked which topological spaces could be represented as the end space of some graph. In 2023, Pitz provided a solution to this question by giving a topological characterization of end spaces using a hereditarily complete special subbase. In this paper, we present an alternative topological characterization of end spaces, in which we employ a special subbase and a topological game. Furthermore, we provide several applications of this characterization: we show that every end space is hereditarily Baire, that $G_δ$ subspaces of end spaces are also end spaces, and that the product of end spaces is not always an end space.
Kui-Yo Chen, Yat-Hin Suen
The conjugacy problem in braid groups has been extensively studied, particularly from an algorithmic perspective. Established methods based on Garside structures, such as initial summit sets and super summit sets, provide effective procedures for determining whether two braids are conjugate. In contrast, explicit structural descriptions of conjugacy classes are less frequently addressed. Although cyclic sliding offers a powerful mechanism for navigating distinguished subsets within a conjugacy class, it is well known that conjugate braids cannot, in general, be obtained from one another solely through iterated cyclic sliding. In this paper, we provide a direct and explicit characterization of the conjugacy classes of positive $3$-braids. Specifically, for any given positive $3$-braid, we determine all of its conjugates in a concrete and closed form.
Benjamin Grant
We define several topological spaces whose points are quivers with a given infinite vertex set $X$. In the special case when $X$ is countably infinite, we show that two of the spaces of interest are homeomorphic to the Baire space $\mathbb{N}^\mathbb{N}$. We study properties of countably infinite quivers as subspaces of these topological spaces and prove a ``meta-theorem'' about hereditary properties of quivers. Furthermore, we approach the question of convergence for infinite mutation sequences in these spaces, providing a complete characterization of the (non-)density of the domains of convergence and divergence of infinite mutation sequences in one of these spaces and a partial characterization in the other. We then draw attention to a very special infinite quiver which we call the \emph{Fraïssé quiver} that draws a clear contrast between the behavior of finite and infinite mutation sequences. Finally, we reproduce (a very mild modification of) a previously-constructed topological space due to Ervin and Jackson as a subquotient of one of the spaces of interest.
Jeremy Brazas, Atish Mitra
In this paper, we study homotopical analogues of the Cantor-Bendixson derivative. For each $n\geq 0$, the "$π_n$-wild set" $\mathbf{w}_n(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. Since the operator $\mathbf{w}_n$ permits iteration, every given space $X$ yields a descending transfinite sequence of nested subspaces $\{\mathbf{w}_n^κ(X)\}_κ$ that stabilizes at some smallest ordinal $\mathbf{wrk}_n(X)$ called the "$π_n$-wild rank" of $X$. We show that the entire transfinite sequence $\{ho(\mathbf{w}_n^κ(X))\}_κ$ of homotopy types is a homotopy invariant of $X$ and that $\mathbf{wrk}_n(X)$ can be an arbitrary countable ordinal when $X$ is an $n$-dimensional Peano continuum. It remains open if there exists a continuum $X$ with uncountable $π_n$-wild rank. This difficulty motivates the parallel study a basepoint-free version $\mathbf{fwrk}_n(X)$, called the "free $π_n$-wild rank" of $X$. We show that for every continuum $X$, $\mathbf{fwrk}_n(X)$ is always countable and can be any countable ordinal.
Haoming Wang
This paper models the theory of abstract harmonic spaces in the syntax of the continuous first-order logic of Banach lattices. It addresses a topological question asking when a one-to-one harmonic map onto smooth manifolds $M^n$ is a diffeomorphism. We give $M^n$ ($n\le 2$) a characterization by $U$-rank and elementary saturation for large cardinals. Polar sets are characterized by several equivalent conditions from the omitting type theorem. Consequently, harmonic measures on the ideal boundary in Martin representation are bijectively mapped to Keisler measures supported on non-principal types. Further problems concerning o-minimality and non-local potentials are finally discussed.
H. S. Behmanush, M. Küçükaslan
In this paper, we introduce the notion of a $γ$-density point for Lebesgue-measurable subsets of $\mathbb{R}$, where $γ$ is a modulus function, and study its basic measure-theoretic properties. We show that every $γ$-density point is a Lebesgue density point, while under Condition~(A) the two notions coincide. Consequently, for such modulus functions, the set of $γ$-density points of a measurable set differs from the set itself only by a null set, yielding a modulus version of the Lebesgue Density Theorem. We then define the associated $γ$-density topology $τ_γ$ and investigate its structure. In general, $τ_γ$ is contained in the classical Lebesgue density topology, and if $γ$ satisfies Condition~(A), then $τ_γ=τ_d$. We also compare $τ_γ$ with $ψ$-density topologies and establish several topological properties of $τ_γ$, including that countable sets are $τ_γ$-closed and that $(\mathbb{R},τ_γ)$ is nonseparable, nonregular, and nonmetrizable. Finally, we introduce $γ$-approximately continuous functions, prove that they form a vector space, and show that the bounded class of such functions is a Banach space under the supremum norm.
Khushbu Gulati, Parameswaran Sankaran
A perforated surface is the complement $\mathringΣ:=Σ\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $Σ$. It is known that the topological type of $Σ\setminus A$ is independent of the choice of $A$. Any perforated surface is one-dimensional, connected, locally path connected, and is not semi-locally simply connected at any of its points. In this paper we obtain a classification theorem for perforated surfaces, using the classification theorem for surfaces. We show that any connected covering of a perforated surface $\mathring Σ$ arises from a covering of a surface $Σ'$ such that $\mathringΣ\cong \mathringΣ'$. We show that the fundamental group of perforated surfaces are large. We also show that the fundamental groups of $\mathring Σ$, the Sierpiński curve and the Menger curve are not Hopfian.
Gabriel Santana, Jemirson Ramirez
Traditional risk measures in finance, predominantly based on the second moment of return distributions or tail risk heuristics (VaR/CVaR), fail to account for the intrinsic geometric structure of market dynamics. This paper introduces a rigorous mathematical framework utilizing Topological Data Analysis (TDA) to quantify risk as the structural instability of the reconstructed phase space. By applying Takens' Delay Embedding Theorem to cryptocurrency log-returns, we generate a point cloud representation of the underlying attractor. We analyze the evolution of the filtration of Vietoris-Rips complexes to compute persistent homology groups $H_k$. We define a "Topological Persistence Norm" to characterize market regimes and propose a leverage calibration heuristic based on the persistence of 1-dimensional cycles. This approach provides a coordinate-free, stability-invariant metric for risk assessment that is robust to high-frequency noise.
Benjamin Vejnar
A fan is an arc-wise connected hereditarily unicoherent continuum with exactly one branching point. By a result of Borsuk, every fan is a 1-dimensional continuum that can be expressed as the union of a family of arcs, each pair of which intersects in the branching point. In this paper, we prove that the converse does not hold by providing a more general result.
Shuhao Li
We study the topology of Lagrangian submanifolds in standard symplectic vector spaces $\mathbb{C}^n$ using ideas from open-closed string topology. Specifically, for a closed, oriented, spin Lagrangian $L$, we construct a (possibly curved) deformation of the dg associative algebra of chains on the based loop space of $L$. This is done via pushing forward moduli spaces of pseudo-holomorphic discs with boundaries on $L$, viewed as chains in the free loop space, along a string topology closed-open map. As an application, we prove that if $π_2(L)=0$, then $L$ has non-vanishing Maslov class, generalizing previous results due to Viterbo, Cieliebak-Mohnke, Fukaya, and Irie.
Ol'ga Sipacheva
It is proved that any countable Boolean topological group has a closed discrete basis.
Jernej Činč, Udayan B. Darji, Benjamin Vejnar
Homeomorphisms of the Cantor set play a central role in topology, dynamical systems and descriptive set theory. In parallel, several classes of fence-like spaces - such as the hairy Cantor set, hairy arcs, Cantor bouquets in complex dynamics, the Lelek fan in topology and Fraïssé fence in descriptive set theory - have recently been studied for their rich structural and dynamical properties. In this paper, we introduce a general construction that associates to each homeomorphism of the Cantor set a canonically defined homeomorphism of a corresponding fence space. This construction lifts dynamical properties from the Cantor set to these fence-like spaces, allowing one to systematically transfer features such as minimality, recurrence, and orbit structure. As a consequence, we obtain a unified framework for studying dynamics on a broad class of fence-like spaces and establish new connections between their topological structure and induced dynamical behavior.
Alexandru Chirvasitu
We classify $n\times n$-matrix-valued continuous commutativity and spectrum preservers defined on spaces of (a) normal, (b) semisimple and (c) arbitrary $n\times n$ matrices with spectra contained in sufficiently connected subsets $\mathcal{X}\subseteq \mathbb{C}$, generalizing a number of results due to Šemrl, Gogić, Tomašević and the author among others. In case (a) these are always conjugations or transpose conjugations, while in cases (b) and (c) qualitatively distinct possibilities arise depending on the local regularity of the complex-conjugation map close to coincident-eigenvalue loci of $\mathcal{X}^n$.
Gerald Kuba
Let c denote the cardinality of the continuum. Let L denote the family of all Hausdorff topologies on the real line coarser than the natural topology. We construct 2^c pairwise non-homeomorphic completely normal topologies in L among which 2^c are Baire and 2^c are of first category. We also construct c pairwise non-homeomorphic completely metrizable topologies in L. Furthermore, we investigate complete lattices of topologies in L and construct extremely long chains of homeomorphic topologies in L.
Judyta Bąk, Joanna Garbulińska-Węgrzyn, Michał Popławski
In this paper, we investigate the set $\mathcal{U}(\mathbb{U})$ of universal and ultrahomogeneous $1$-Lipschitz retractions acting on the Urysohn space as the subspace of the space $\mathcal{R}(\mathbb{U})$ of all $1-$Lipschitz retractions defined on the Urysohn space. Especially, we study Borel complexity and density $\mathcal{U}(\mathbb{U})$ in $\mathcal{R}(\mathbb{U}).$ In order to do that, we introduce a new extension property $(UR^*)$ that is equivalent to the universality and ultrahomogeneity of a retraction, and a new pointwise retract topology.